AUTOBIOGRAPHY.

I Was born in Chicago on the twenty-fourth day of October, 1879 After completing the course in the primary and secondary schools of the City of Chicago, I entered the College of Liberal Arts of Northwestern University in September, 1896. I was graduated in June, 1900, receiving the degree Bachelor of Arts. The next two years I spent in graduate work in Mathematics and Physics at Harvard University, and received the degree Master of Arts in June, 1902. I continued my work for two years in the Graduate School of the University of Chicago as a Fellow in Mathematics.

I wish to acknowledge my indebtedness to all of the instructors under whose direction I have worked as a graduate student, and to express my appreciation of the kindly interest manifested by Professor Dickson, under whose direction this investigation has been carried out.

WILLIAM HENRY BUSSEY.

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SIMPLY ISOMORPHIC WITH THE GROUP LF[2,pn]

By W. H. Busset.
Communicated by Prof. L. E. Dickson.

[Received December 10th, 1904.—Read January 12th, 1905.]

[Extracted from the Proceedings of the London Mathematical Society, Ser. 2, Vol. 3, Part 4.]

Introduction.

1. The object of this paper is the proof of the following six theorems concerning sets of generational relations for the abstract group G, simply isomorphic with the group LF [2, pn~] of all linear fractional transformations, on one variable, having determinant unity and coefficients belonging to the GF(pn).

Theorem I.*—The abstract group GjP(p>_i), simply isomorphic with the group LF [2, p], p > 2, may be generated by two operators T and S, subject to the generational relations

(A) &> = I, T2 = /, (ST)8 = I, (SrTS2TT)2 = I, T 0.

Theorem II.—The abstract group G^^^, simply isomorphic with the group LF[2, p*], p > 2, n > 1, may be generated by (pn-f-l) operators T and Sx, X running through the marks of the GF(pn), subject to the generational relations

(1) S0 = I, SxSp = S*+Il (X, n any marks),

(2) T*=I, (S1Tf = I,

(J3)

(8) (S» TSs/r Tf = I (t any mark =f= 0),

,(4) [1/a, a2], [1/a, ia4], [t, a], a] (a =£ 0),

where i is a primitive root of the GF(pn), and a is any mark subject to a restriction implied in the notation [X, ft].

* For the special cases in which pn < 47, this theorem has been proved by Prof. Dickson, Proc. London Math. Soc., Vol. xxxv., pp. 292-305; Bull. Amer. Math. Soc, Vol. ix., p. 297.

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Note.—The symbol [X, M] is used to denote the relation*

where X, n are marks such that X/x 1.

Theorem III. t—For the special cases in which pn = 49, 81, 121 relations (1), (2), (8) form a set of generational relations for G^^-v), P > 2.

Theorem IV. t—The abstract group G2,.(2-2»-i), simply isomorphic with the group LF[2, 2n], may be generated by three operators a, b, c, subject to the generational relations

(5) a2n-1 = I, 62 = J, ba(b = a"6af,

(C)

| (6) c* = I, (ca)a = /, (cb)3 = I,

where £ = 1, 2, 8, (2n—2), and i?, £ are determined by the relations i( l+i(, >] = £—£, mod (2n—1), i being a primitive root of the GF(2n).

Theorem V.—The abstract group G2n(2s»_i), simply isomorphic with the group LF{2, 2n], may be generated by two operators a and d subject to the generational relations

CD) a2n^ -I, d* = I, (da(d^atf = I, (datda()e = I,

where £ = 1, 2, 8 (2n—2), and £ is determined by the relation

is = l+if, i being a primitive root of the GF(2n).

Theorem VI.§—In the special cases in which n = 2, 8, 4, 5, 6, the abstract group G2»(2jb_1), simply isomorphic with the group LF [2, 2n],

* Relations (1), T* = I, [A, pi], A, n any marks suoh that A^=j£ 1, constitute a set of generational relations for 0. This is a special case of a more general theorem valid for any field due to Moore. See Proc. London Math. Soc, Vol. xxxv., p. 293, and Dickson's Linear Groups, p. 300.

Note that, when \ - 0 or 1, [A, n] reduces to (5, T)% = /, and, when A = — 1, [A, p] reduces to (3).

t For the special cases in which pn = 9, 25, 2", 125, 243, Prof. Dickson has proved that (1), (2), (3) constitute a set of generational relations for ^jj,'>(p!'>-i), P> ^> ^e proof"

for the cases in which pn = 125, 243 have not been published.

X This theorem is due to de Seguier, Journal de Mathcmatiques, Tome vm., p. 253.

§ The set of generational relations (E) is due to Prof. Dickson. He has proved Theorem VI. for n = 2, 3, 4. See Proc. London Math. Soc, Vol. xxxv., p. 306 and p. 443; Bull. Amur. Math. Soc., Vol. ix., pp. 194-204.

For n = 2, the set (E) reduces to Ah = /, B1 = /, (ABf = J.

may be generated by two operators A and B subject to the generational relations

(E) Ar+1 = I, B2 = I, (AB)a = I, (BArBAf = I,

where r = 1, 2, 8, 2n, and the value of s is determined by the relation f(f+ts+l) = P(**+1)+1, toeing defined by the relation f = t*f+l, i being a primitive root of the GF(1n).

The Group G Simply Isomorphic With LF [2, pn\ p > 2.

2. Lemma.The abstract group G^pK(Jp>_Y), simply isomorphic with the group LF[2, pn], p > 2, may be generated by (pn+2) operators R, T, and Sx, X running through the marks of GF(pn), subject to relations (1) and

(7) fl*"n^) = I,

(8) SxR' = R'Stf* (X any mark, a- = 0 or any integer),

(9) (TR')* = I (a- = 0 or any integer),

(10) TSyT = R"S-yTS-yy (y any mark =£0, if = —y),

i being a primitive root of the GF(pn).

Proof. — The group LF[2, pn~\, p>%, may be generated by the pn+l transformations

T :V = —, Sx : z' = z+\ (X any mark),

z

while the sub-group K of the transformations

[graphic]

may be generated by the transformations

S1.X = SA : J z+\, B:z' = izli-\

i being a primitive root of the GF(pn).

These generators of K satisfy relations (1), (7), and (8). Since the group LF [2, p"\ p > 2, when represented as a permutation group on (pn-\-l) letters, is doubly transitive while the sub-group K, being then a permutation group on pn letters, is simply transitive, it follows from the work of Jordan* that it is possible to determine y, S, 17, f, and u such that

"Traite des Substitution*, p. 32.

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